We present a new algorithm, the COLD PRISM, for computing matrix elements in molecular orbital calculations. Whereas traditional approaches form these from two-electron repulsion integrals (ERIs) which, in turn, are formed from shell-pair data, we introduce several alternative paths that do not involve ERIs as intermediates. Tensor multiplication can be used as the basic arithmetic operation on all of the new PRISM paths and the associativity of tensor multiplication plays a key role. We have implemented our approach in the Q-Chem program.